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Discrete Homotopy and subspace arrangements

Algebra & Discrete Mathematics

Speaker: Helene Barcelo, MSRI
Location: 2112 MSB
Start time: Fri, Dec 5 2008, 2:10PM

In 1962, Fadell and Neuwirth showed that removing all the diagonals $z_i=z_j$ from a complex $n$-dimensional space yield a $K(\pi, 1)$ space with fundamental group isomorphic to the pure braid group.

In 1996, Khovanov proved a real counterpart to this theorem. That is, starting with a real $n$-dimensional space and removing all real codimension two subspaces of the form $x_i=x_j=x_k$ yields a $K(\pi,1)$ space. The set of all real subspaces of the form $x_{i_1}=x_{i_2}= ... =x_{i_k}$ is called the $k$-equal arrangement; Khovanov also showed that for all $k \geq 4$, the complement of a $k$-equal arrangement has only trivial homotopy groups.

We generalize $k$-equal arrangements to $k$-parabolic arrangements and study the corresponding complements, $M_k(W)$, where $W$ is a (real) finite Coxeter group. We show that the fundamental group of $M_k(W)$ is isomorphic to the discrete fundamental group $A_1^{n-k}$ of the Coxeter complex associated to $W$, generalizing the independent results of Babson and Bjorner for the type $A$ case. We use this result to show that given two Coxeter groups, the fundamental group of $M_k(W_1 \times W_2)$ is isomorphic to the direct product of the corresponding fundamental groups of $M_k(W_1)$ and $M_k(W_2)$. Finally we generalize a result of Khovanov showing that the fundamental homotopy group of $M_2(W)$ is a normal subgroup of an infinite Coxeter group of index $|W|$. We conjecure that several of these results hold for higher homotopy groups.